Question

The point (-7, -24) is on the terminal ray of angle θ which is in standard position. A student found the six trigonometric values for angle θ. The student’s answers are shown

Answer 1

$\bf sin(\theta)=\cfrac{opposite}{hypotenuse} \qquad cos(\theta)=\cfrac{adjacent}{hypotenuse} \quad % tangent tan(\theta)=\cfrac{opposite}{adjacent} \\\\\\ % cotangent cot(\theta)=\cfrac{adjacent}{opposite} \qquad % cosecant csc(\theta)=\cfrac{hypotenuse}{opposite} \quad % secant sec(\theta)=\cfrac{hypotenuse}{adjacent}\\\\ -------------------------------$

$\bf (\stackrel{a}{-7}~~,~~\stackrel{b}{-24}) \\\\\\ \textit{using the pythagorean theorem}\\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2}\qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c=\sqrt{(-7)^2+(-24)^2}\implies c=\sqrt{49+576}\implies c=\sqrt{625} \\\\\\ c=25\impliedby hypotenuse\\\\ -------------------------------$

$\bf sin(\theta )=\cfrac{-24}{25}\qquad cos(\theta )=\cfrac{-7}{25}\qquad \qquad tan(\theta )=\cfrac{-24}{-7}\implies \cfrac{24}{7} \\\\\\ cot(\theta )=\cfrac{-7}{-24}\implies \cfrac{7}{24}\qquad \qquad sec(\theta )=\cfrac{25}{-7}\qquad csc(\theta )=\cfrac{25}{-24}$

Answer 2

The student wrongly calculated the value of $\rm tan\theta$, $\rm sin\theta$, $\rm cosec\theta$, and $\rm cot\theta$ and this can be determined by using the Pythagorean theorem and trigonometric functions.

GIven :

The point (-7, -24) is on the terminal ray of angle θ which is in standard position.

To determine the trigonometric function first determine the value of the hypotenuse and to determine the value of the hypotenuse, the Pythagorean theorem can be used.

$\rm H^2 = P^2 + B^2$

where H is the hypotenuse, B is the base, and P is the perpendicular.

Now, substitute the value of known terms in the above equation.

$\rm H^2 = 7^2+24^2$

$\rm H^2=49+576$

$\rm H^2 = 625$

H = 25

Now, the values of the trigonometric functions are as follows:

$\rm cos\theta=\dfrac{-7}{25}$

$\rm sin\theta=\dfrac{-24}{25}$

$\rm tan\theta=\dfrac{24}{7}$

$\rm cosec\theta=\dfrac{25}{-24}$

$\rm sec\theta=\dfrac{25}{-7}$

$\rm cot\theta=\dfrac{7}{24}$

The student wrongly calculated the value of $\rm tan\theta$, $\rm sin\theta$, $\rm cosec\theta$, and $\rm cot\theta$.

For more information, refer to the link given below:

https://brainly.com/question/21286835